Save text logo for external use
Using a non-animated image as a logo/overlay in an external program is generally pretty simple. From blender, all you should need to do it render it as an image with a transparent background. It should then be easy to add it to your program of choice. For example in OBS i found this: https://obsproject.com/forum/threads/inserting-a-logo-while-recording-a-video.19133/

For more info on rendering a transparent background see: Can Blender render pngs with the background transparent?